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+Symbolics with Sage
+-------------------
+
+Hello friends and welcome to this tutorial on symbolics with sage.
+
+
+.. #[Madhu: Sounds more or less like an ad!]
+
+{{{ Part of Notebook with title }}}
+
+.. #[Madhu: Please make your instructions, instructional. While
+     recording if I have to read this, think what you are actually
+     meaning it will take a lot of time]
+
+We would be using simple mathematical functions on the sage notebook
+for this tutorial.
+
+.. #[Madhu: What is this line doing here. I don't see much use of it]
+
+During the course of the tutorial we will learn
+
+{{{ Part of Notebook with outline }}}
+
+To define symbolic expressions in sage.  Use built-in costants and
+function. Integration, differentiation using sage. Defining
+matrices. Defining Symbolic functions. Simplifying and solving
+symbolic expressions and functions.
+
+.. #[Nishanth]: The formatting is all messed up
+                First fix the formatting and compile the rst
+                The I shall review
+.. #[Madhu: Please make the above items full english sentences, not
+     the slides like points. The person recording should be able to
+     read your script as is. It can read something like "we will learn
+     how to define symbolic expressions in Sage, using built-in ..."]
+
+Using sage we can perform mathematical operations on symbols.
+
+.. #[Madhu: Same mistake with period symbols! Please get the
+     punctuation right. Also you may have to rephrase the above
+     sentence as "We can use Sage to perform sybmolic mathematical
+     operations" or such]
+
+On the sage notebook type::
+   
+    sin(y)
+
+It raises a name error saying that y is not defined. But in sage we
+can declare y as a symbol using var function.
+
+.. #[Madhu: But is not required]
+::
+    var('y')
+   
+Now if you type::
+
+    sin(y)
+
+    sage simply returns the expression .
+
+.. #[Madhu: Why is this line indented? Also full stop. When will you
+     learn? Yes we can correct you. But corrections are for you to
+     learn. If you don't learn from your mistakes, I don't know what
+     to say]
+
+thus now sage treats sin(y) as a symbolic expression . You can use
+this to do a lot of symbolic maths using sage's built-in constants and
+expressions .
+
+.. #[Madhu: "Thus now"? It sounds like Dus and Nou, i.e 10 and 9 in
+     Hindi! Full stop again. "a lot" doesn't mean anything until you
+     quantify it or give examples.]
+
+Try out
+
+.. #[Madhu: "So let us try" sounds better]
+ ::
+   
+   var('x,alpha,y,beta') x^2/alpha^2+y^2/beta^2
+ 
+Similarly , we can define many algebraic and trigonometric expressions
+using sage .
+
+.. #[Madhu: comma again. Show some more examples?]
+
+
+Sage also provides a few built-in constants which are commonly used in
+mathematics .
+
+example : pi,e,oo , Function n gives the numerical values of all these
+    constants.
+
+.. #[Madhu: This doesn't sound like scripts. How will I read this
+     while recording. Also if I were recording I would have read your
+     third constant as Oh-Oh i.e. double O. It took me at least 30
+     seconds to figure out it is infinity]
+
+For instance::
+
+   n(e)
+   
+   2.71828182845905
+
+gives numerical value of e.
+
+If you look into the documentation of n by doing
+
+.. #[Madhu: "documentation of the function "n"?]
+
+::
+   n(<Tab>
+
+You will see what all arguments it can take etc .. It will be very
+helpful if you look at the documentation of all functions introduced
+
+.. #[Madhu: What does etc .. mean in a script?]
+
+Also we can define the no of digits we wish to use in the numerical
+value . For this we have to pass an argument digits.  Type
+
+.. #[Madhu: "no of digits"? Also "We wish to obtain" than "we wish to
+     use"?]
+::
+
+   n(pi, digits = 10)
+
+Apart from the constants sage also has a lot of builtin functions like
+sin,cos,sinh,cosh,log,factorial,gamma,exp,arcsin,arccos,arctan etc ...
+lets try some out on the sage notebook.
+
+.. #[Madhu: Here "a lot" makes sense]
+::
+     
+   sin(pi/2)
+   
+   arctan(oo)
+     
+   log(e,e)
+
+
+Given that we have defined variables like x,y etc .. , We can define
+an arbitrary function with desired name in the following way.::
+
+       var('x') function(<tab> {{{ Just to show the documentation
+       extend this line }}} function('f',x)
+
+.. #[Madhu: What will the person recording show in the documentation
+     without a script for it? Please don't assume recorder can cook up
+     things while recording. It is impractical]
+
+Here f is the name of the function and x is the independent variable .
+Now we can define f(x) to be ::
+
+     f(x) = x/2 + sin(x)
+
+Evaluating this function f for the value x=pi returns pi/2.::
+	   
+	   f(pi)
+
+We can also define functions that are not continuous but defined
+piecewise.  We will be using a function which is a parabola between 0
+to 1 and a constant from 1 to 2 .  type the following as given on the
+screen
+
+.. #[Madhu: Instead of "We will be using ..." how about "Let us define
+     a function ..."]
+::
+      
+
+      var('x') h(x)=x^2 g(x)=1 f=Piecewise(<Tab> {{{ Just to show the
+      documentation extend this line }}}
+      f=Piecewise([[(0,1),h(x)],[(1,2),g(x)]],x) f
+
+Checking f at 0.4, 1.4 and 3 :: f(0.4) f(1.4) f(3)
+
+.. #[Madhu: Again this doesn't sound like a script]
+
+for f(3) it raises a value not defined in domain error .
+
+
+Apart from operations on expressions and functions one can also use
+them for series .
+
+.. #[Madhu: I am not able to understand this line. "Use them as
+.. series". Use what as series?]
+
+We first define a function f(n) in the way discussed above.::
+
+   var('n') function('f', n)
+
+.. #[Madhu: Shouldn't this be on 2 separate lines?]
+
+To sum the function for a range of discrete values of n, we use the
+sage function sum.
+
+For a convergent series , f(n)=1/n^2 we can say ::
+   
+   var('n') function('f', n)
+
+   f(n) = 1/n^2
+
+   sum(f(n), n, 1, oo)
+
+For the famous Madhava series :: var('n') function('f', n)
+
+.. #[Madhu: What is this? your double colon says it must be code block
+     but where is the indentation and other things. How will the
+     recorder know about it?]
+
+    f(n) = (-1)^(n-1)*1/(2*n - 1)
+
+This series converges to pi/4. It was used by ancient Indians to
+interpret pi.
+
+.. #[Madhu: I am losing the context. Please add something to bring
+     this thing to the context]
+
+For a divergent series, sum would raise a an error 'Sum is
+divergent' :: 
+	
+	var('n') 
+	function('f', n) 
+	f(n) = 1/n sum(f(n), n,1, oo)
+
+
+
+
+We can perform simple calculus operation using sage
+
+.. #[Madhu: When you switch to irrelevant topics make sure you use
+    some connectors in English like "Moving on let us see how to
+    perform simple calculus operations using Sage" or something like
+    that]
+For example lets try an expression first ::
+
+    diff(x**2+sin(x),x) 2x+cos(x)
+
+The diff function differentiates an expression or a function . Its
+first argument is expression or function and second argument is the
+independent variable .
+
+.. #[Madhu: Full stop, Full stop, Full stop]
+
+We have already tried an expression now lets try a function ::
+
+   f=exp(x^2)+arcsin(x) diff(f(x),x)
+
+To get a higher order differentiation we need to add an extra argument
+for order ::
+ 
+   diff(<tab> diff(f(x),x,3)
+
+.. #[Madhu: Please try to be more explicit saying third argument]
+
+in this case it is 3.
+
+
+Just like differentiation of expression you can also integrate them ::
+
+     x = var('x') s = integral(1/(1 + (tan(x))**2),x) s
+
+.. #[Madhu: Two separate lines.]
+
+To find the factors of an expression use the "factor" function
+
+.. #[Madhu: See the diff]
+
+::
+    factor(<tab> y = (x^100 - x^70)*(cos(x)^2 + cos(x)^2*tan(x)^2) f =
+    factor(y)
+
+One can also simplify complicated expression using sage ::
+    f.simplify_full()
+
+This simplifies the expression fully . You can also do simplification
+of just the algebraic part and the trigonometric part ::
+
+    f.simplify_exp() f.simplify_trig()
+    
+.. #[Madhu: Separate lines?]
+
+One can also find roots of an equation by using find_root function::
+
+    phi = var('phi') find_root(cos(phi)==sin(phi),0,pi/2)
+
+.. #[Madhu: Separate lines?]
+
+Lets substitute this solution into the equation and see we were
+correct ::
+
+     var('phi') f(phi)=cos(phi)-sin(phi)
+     root=find_root(f(phi)==0,0,pi/2) f.substitute(phi=root)
+
+.. #[Madhu: Separate lines?]
+
+as we can see the solution is almost equal to zero .
+
+.. #[Madhu: So what?]
+
+We can also define symbolic matrices ::
+
+
+
+   var('a,b,c,d') A=matrix([[a,1,0],[0,b,0],[0,c,d]]) A
+
+.. #[Madhu: Why don't you break the lines?]
+
+Now lets do some of the matrix operations on this matrix
+
+.. #[Madhu: Why don't you break the lines? Also how do you connect
+     this up? Use some transformation keywords in English]
+::
+    A.det() A.inverse()
+
+.. #[Madhu: Why don't you break the lines?]
+
+You can do ::
+    
+    A.<Tab>
+
+To see what all operations are available
+
+.. #[Madhu: Sounds very abrupt]
+
+{{{ Part of the notebook with summary }}}
+
+So in this tutorial we learnt how to
+
+
+We learnt about defining symbolic expression and functions .  
+And some built-in constants and functions .  
+Getting value of built-in constants using n function.  
+Using Tab to see the documentation.  
+Also we learnt how to sum a series using sum function.  
+diff() and integrate() for calculus operations .  
+Finding roots , factors and simplifying expression using find_root(), 
+factor() , simplify_full, simplify_exp , simplify_trig .
+Substituting values in expression using substitute function.
+And finally creating symbolic matrices and performing operation on them .
+
+.. #[Madhu: See what Nishanth is doing. He has written this as
+     points. So easy to read out while recording. You may want to
+     reorganize like that]